I have a week and will start in Paris...

Oh-boy, a week in France!
Literally hundreds of places to go.

From your email:
"love wine" - you can get good wine most placesin France, but if you want to visit wine production regions, there would be Bordeaux and Burgundy (and many other wine production regions); Reims for champagne, little over an hour from Paris on the fast train.

"love food" - again, many places, including Paris offer great food, but if you want to go someplace other than Paris, about anywhere of any size and he Michelin RED Guide (available online) is just one source for good restaurants. Dijon is a special foodie place.

"love art" - you're probably already aware of the many Paris art museums (Orsay, Louvre, and many more); Arles & Saint Remy in southern France for Van Gogh locations in the latter 2 yrs of his life, (but not his original paintings, those are mostly in Amsterdam and the Kroller-Mueller an hour east of Amsterdam, although the Orsay has about a room of them (about 15) but you can see around three hundred of his paintings at the Van Gogh Museum (Amsterdam) and the Kroller-Mueller).

Everyone loves food. Go to the Provence area,. Just take the TGV and you are down there in 3 hours or less. Stay by Arles or Nimes. Go see the Pont du Gard and the other Roman stuff in the area. Go to Avignon. Get a tour from there 4 hour or 8 hour to see villages like Gordes, Rousillion, Les Baux. Include a tour to Chateauneuf de Pape winery from Avignon. You can arrange all of this at the Tourist Info Center of most major towns in Provence.

Your problem with southern France destinations might be the travel time involved in getting from/to Paris, it will eat some into your available time--although there are some pretty fast trains on the route from Paris directly south to various destinations.

And lots of places much closer to Paris than southern France, such as Normandy, Chartres, and the Loire Valley.

You probably already have guidebooks or online resources for what's in France and where.

You mentioned budgetary considerations, you could do day trips from Paris but then you'd still be paying Paris lodging prices.

Thanks for the great suggestions! I have only been in and around Paris in France (Versailles, Giverny) so I think leaving the city would be a good thing! I am open to leaving France as well, but with only a week, I wasn't sure what was feasible. I am looking for a fairly relaxing adventure, as I will have just been on an exhausting trip with high school students.

Would it make the most sense to use trains to get around? Provence sounds fabulous. Too early for lavender?
Should I venture down to to coast?

The Lavender blooms in late June. fly to Nice it only takes 1 hour 30 minutes if you want to go to the coast. They have all the art down there Matisse Musuem, Picasso (the house he lived in at Antibes is a museum on the coast if the Mediterranean, St Paul De Vence is full of galleries etc. Just google that.

The sooner you decide the less it will cost you at this point.

Regarding your question about transportation: Yes, you could do the long distances by train, supplemented as needed by van day tour or rental car, even bicycle.

You asked about "the coast."
Which coast did you mean, France has 3 of them.
Do you have a map, if not, you're going to want to take a look at one and the hard copy maps are easiest to use.

With European budget airlines, you could pick just about any city on the continent that draws your interest.
Sounds as if wine, food and art would draw you to Italy.
I'd be going back to Budapest.

A week on your own, starting in Paris? I'd go to one of these three in June, Burgundy, Alsace or Champagne (in that order).

Burgundy has the best wine on the planet. You could train to Auxerre, rent a car and wander down to Beune/Dijon (I personally prefer Dijon's center though it's a larger city) via some amazingly nice medieval towns, farmland and forests. Train back to Paris from either Beune or Dijon.

Alsace is compact and has terrific dry white wines (best is Gewurztraminer). Train to Colmar, visit some of the small towns along the wine road (like Burgundy each is prettier than the last). Visit a great castle, Chateau du Haut-Koenigsbourg, then end in Strasbourg and train back to Paris.

Champagne is a place I haven't really explored. Reims (pronounced something like Hrance, don't ask me why) is a great visit for Champagne houses, great cathedral and some Roman ruins. I also like the mostly pedestrian restaurant zone near the train station. Troyes has a big half-timbered medieval center. Epernay is maybe the third major place to visit. It's very easy to train from Paris to Reims as a day trip, or you could stop there on the way back from Strasbourg as I did.

I was able to find Airbnb accommodations for roughly 50 euro per night all over Burgundy and Alsace, especially when you aren't in high season (July/August).

For me, a fellow wine lover, from Paris, the obvious choices would be Loire, Champagne, Burgundy and Alsace. Consider Loire if you are willing to rent a car and see some chateaus and sample wines. Consider Alsace if you like adorable little villages and miles of grapes. Consider Burgundy (Beaune/Dijon) if you like world class wine and more of a medieval look. The route de grand crus from Beaune to Dijon is beautiful. I haven't been to Champagne. For me, my favorite is Alsace. You could consider from Paris stopping in Champagne and then continuing on to Alsace. You could take a train from Champagne to Colmar. Colmar is nice, but the real gems are the small villages more easily visited by car. If you don't mind switching accommodations (I don't) you could stop a night in Strausbourg before heading on to Colmar. Both Strasbourg and Colmar can be visited without a car.